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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.8.27
Chapter 4, Problem 4.8.27

{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.


y = sin x and y = x/2

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Step 1: Set the equations equal to each other to find the intersection points. This means solving sin(x) = x/2.
Step 2: Rearrange the equation to form f(x) = sin(x) - x/2 = 0. This is the function whose roots we want to find using Newton's method.
Step 3: Compute the derivative of f(x), which is f'(x) = cos(x) - 1/2. This derivative will be used in Newton's method.
Step 4: Choose initial approximations for x based on preliminary graphing or analysis. Graphing the functions y = sin(x) and y = x/2 can help identify where they might intersect.
Step 5: Apply Newton's method iteratively using the formula x_{n+1} = x_n - f(x_n)/f'(x_n) until the values converge to a satisfactory level of precision for each intersection point.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Method

Newton's Method is an iterative numerical technique used to find approximate solutions to equations. It starts with an initial guess and refines it using the formula x_{n+1} = x_n - f(x_n)/f'(x_n), where f is the function and f' is its derivative. This method is particularly useful for finding roots of functions, making it ideal for determining intersection points of curves.
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Intersection Points

Intersection points occur where two curves meet, meaning their y-values are equal for the same x-value. To find these points, one typically sets the equations of the curves equal to each other and solves for x. In this case, finding the intersection of y = sin x and y = x/2 involves solving sin x = x/2, which may not have a straightforward algebraic solution.
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Graphical Analysis

Graphical analysis involves plotting functions to visually identify their behavior and potential intersection points. By graphing y = sin x and y = x/2, one can observe where the curves intersect, which aids in selecting appropriate initial guesses for Newton's Method. This visual approach can enhance understanding of the functions' dynamics and the nature of their intersections.
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Related Practice
Textbook Question

Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>

Textbook Question

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.


ƒ(x) = 2 sinx + 1

Textbook Question

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f(x) = 4x² + x; a = 1

Textbook Question

Covering a marble Imagine a flat-bottomed cylindrical pot with a circular cross section of radius 4. A marble with radius 0 < r < 4 is placed in the bottom of the pot. What is the radius of the marble that requires the most water to cover it completely?

Textbook Question

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = 2 - a cos x, a constant

Textbook Question

Mean Value Theorem and the police A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to a patrol officer 30 mi along the highway. When the car reached the location of the second officer 28 min later, it was clocked going 60 mi/hr. The driver of the car was given a ticket for exceeding the 60-mi/hr speed limit. Why can the officer conclude that the driver exceeded the speed limit?